On the Hughes--Keating--O’Connell Conjecture: \\ Quantified Negative Moment Bounds for $\zeta'(\rho)$ via Entropy--Sieve Methods Revisited

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We study negative discrete moments of the derivative of the Riemann zeta function at its nontrivial zeros. Using a novel \textbf{entropy--sieve method (ESM)}, and assuming the Riemann Hypothesis together with mild pair-correlation and discrete moment hypotheses, we establish a \textbf{quantified conditional bound} on negative moments: \[ J_{-1}(T)\le C(\varepsilon)\,T(\log T)^{\varepsilon}. \] Our approach combines Dirichlet polynomial approximations, Gaussian cumulant estimates, and a small-gap sieve. This framework matches the conjectured asymptotics up to logarithmic factors and has implications for the \textbf{simplicity of zeros}.

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